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Safety issues for the use of products containing nanoparticles need to be considered, since these nanoparticles may break through human skin to damage cells. In this paper, applied mathematical techniques are used to model the penetration of a spherical gold nanoparticle into an assumed circular hole in a lipid bilayer. The 6–12 Lennard-Jones potential is employed, and the total molecular interaction energy is obtained using the continuous approximation. Nanoparticles of three different radii, namely, 10, 15 and 20 Å, are studied, which are initiated at rest, confined to the axis of the hole. A similar behaviour for these three cases is observed. The critical hole radii at which these nanoparticles enter the bilayer are 12.65, 17.62 and 22.60 Å, respectively. Further, once the hole radii become larger than 20.79, 23.14 and 27.02 Å, respectively, the gold nanoparticles tend to remain at the mid-plane of the bilayer, and do not pass through the bilayer. [ABSTRACT FROM AUTHOR]

We review the work of the present authors to employ variational calculus to formulate continuous models for the connections between various carbon nanostructures. In formulating such a variational principle, there is some evidence that carbon nanotubes deform as in perfect elasticity, and rather like the elastica, and therefore we seek to minimize the elastic energy. The calculus of variations is utilized to minimize the curvature subject to a length constraint, to obtain an Euler–Lagrange equation, which determines the connection between two carbon nanostructures. Moreover, a numerical solution is proposed to determine the geometric parameters for the connected structures. Throughout this review, we assume that the defects on the nanostructures are axially symmetric and that the into-the-plane curvature is small in comparison to that in the two-dimensional plane, so that the problems can be considered in the two-dimensional plane. Since the curvature can be both positive and negative, depending on the gap between the two nanostructures, two distinct cases are examined, which are subsequently shown to smoothly connect to each other. [ABSTRACT FROM AUTHOR]